15.7 Triple integrals in Cylindrical Coordinates

Claudia Castro-Castro
Math 283 Spring 2020

Instructions:

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Outline

  • The following topics will be covered in this lecture:
    • Cylindrical coordinates
    • Volume element in cylindrical coordinates
    • Evaluation of triple integrals using cylindrical coordinates

Triple Integrals in Cylindrical Coordinates

  • In 3D dimensions there is a coordinate system, called cylindrical coordinates, that is similar to polar coordinates and gives convenient descriptions of some commonly occurring surfaces and solids
  • Suppose you want to find the total mass of the half cylinder with density function \( f(x,y,z) \)
Diagram of polar coordinates

Courtesy of James Stewart, Calculus: Early transcendentals, 2nd edition

  • Suppose you want to find the total mass of the half cylinder with density function \( \rho(x,y,z) \) \[ mass=\iiint \limits_{\frac{1}{2} Cylinder} \rho(x,y,z)dV \]
  • Projections

Recall: polar coordinates

  • Cartesian coordinates \( (x, y) \) and polar coordinates \( (r,\theta) \)
Diagram of polar coordinates

Courtesy of James Stewart, Calculus: Early transcendentals, 2nd edition

  • Polar to rectangular \( (r,\theta)\rightarrow (x,y) \)
    \[ x=r\cos \theta \\y=r\sin \theta \]

  • Rectangular to polar \( (x,y)\rightarrow (r,\theta) \)
    \[ r^2=x^2+y^2 \\ \tan \theta = \frac{y}{x},\;\cos \theta = \frac{x}{r},\;\sin \theta = \frac{y}{r} \]

Cylindrical coordinates

Diagram of polar coordinates

Courtesy of James Stewart, Calculus: Early transcendentals, 2nd edition

  • A point \( P \) in 3-D space is represented by the ordered triple \[ (r, \theta , z) \]
  • where \( r \) and \( \theta \) are polar coordinates of the projection of \( P \) onto the \( xy \)-plane
  • and \( z \) is the directed distance from the \( xy \) -plane to \( P \)
  • Cylindrical to rectangular \( (r,\theta,z)\rightarrow (x,y,z) \)
    \[ \begin{align} x=&r\cos \theta \\y=&r\sin \theta\\z=&z \end{align} \]

  • Rectangular to cylindrical \( (x,y,z)\rightarrow (r,\theta,z) \)
    \[ r=\sqrt{x^2+y^2} \\ \tan \theta = \frac{y}{x},\;\cos \theta = \frac{x}{r},\;\sin \theta = \frac{y}{r}\\z=z \]

Points in Cylindrical coordinates

Example:

  • Find cylindrical coordinates of the point with rectangular coordinates \( (x, y, z) = (-1, 1, 3) \)
  • Want \( (r,\theta, z) \)
  • \( z \) is the same as in Rectangular coords \( z=3 \)
  • Use parametric equations \[ r=\sqrt{x^2+y^2} \]
  • \[ r=\sqrt{(-1)^2+(1)^2} = \sqrt{2} \]
  • \[ \tan\theta = \frac{y}{x}= \frac{1}{-1}= -1 \]
  • or \[ \cos\theta = \frac{x}{r}=-\frac{1}{\sqrt{2}}= \frac{\sqrt{2}}{2} \]
  • \[ \Leftrightarrow \theta = \frac{3}{4}\pi \]

  • Therefore \( (r,\theta, z) = \left(\sqrt{2}, \frac{3}{4}\pi, 3\right) \)

Solids in Cylindrical coordinates

  • We can re-define certain solids in Cylindrical coordinates.

Example: Cylinder

  • The cyircular cylinder with Cartesian equation \[ x^2+y^2=c^2 \]

  • In cylindrical coordinates the surface of this cylinder has the equation \[ r=c \]
  • and the solid cylinder as the set
    \[ E=\left\{(r,\theta,z)|\; 0 \leq r\leq c,\; 0 \leq \theta \leq 2\pi,\; a\leq z\leq b \right\} \]
Circular cylinder of radius c

Courtesy of James Stewart, Calculus: Early transcendentals, 2nd edition

Solids in Cylindrical coordinates

Example: Cone

  • The cyircular cone with Cartesian equation \[ z^2=x^2+y^2 \]

  • Cone has radial symmetry
  • In cylindrical coordinates the surface of this cone has the equation \[ z=r \]
  • and the solid cone as the set
    \[ E=\{(r,\theta,z)|\; 0 \leq r\leq c,\; 0 \leq \theta \leq 2\pi,\; r \leq z\leq s \} \]
Circular cone of radius c

Courtesy of James Stewart, Calculus: Early transcendentals, 2nd edition

Evaluating Triple integrals in Cylindrical Coordinates

Generic solid in cylindrical coordinates

Courtesy of James Stewart, Calculus: Early transcendentals, 2nd edition

  • Suppose that \( E \) is a type 1 solid with projection \( D \) onto, say, the \( xy \)-plane is conveniently described in polar coordinates
  • We know that \[ \iiint_\limits{E} f(x,y,z)dV=\iint_\limits{D}\left[ \int_{\color{purple} {u_1(x,y)}}^{\color{purple} {u_2(x,y)}} f(x,y,z) {\color{purple} {dz}} \right] dA \]
  • \( D \) is given in polar coordinates
  • Convert a triple integral from rectangular to cylindrical coordinates using \[ \begin{align} x&=r\cos\theta\\ y&=r\sin\theta\\ z&=z\end{align} \]
  • \[ \iiint_\limits{E} f(r,\theta,z)dV_{cyl} = \iint_\limits{D}\left[ \int_{{\color{purple} {u_1(r,\theta)}}}^{{\color{purple} {u_2(r,\theta)}}} f(r,\theta,z) dz \right] dA_{polar} \]

  • What is \( dV_{cyl} \)?

Volume element in Cylindrical Coordinates

Volume element in cylindrical coords

Courtesy of James Stewart, Calculus: Early transcendentals, 2nd edition

  • Volume element in cylindrical coordinates has volume

    \[ dV_{cyl} =(Area\; of\; polar\;rectangle)(height) \]
  • \[ dV_{cyl} =(dA_{polar})(dz) \]
  • \[ dV_{cyl} =(rdrd\theta)(dz) \]
  • \[ dV_{cyl} = r\: dz\: dr\:d\theta \]

Example

  • A solid \( E \) lies within the cylinder \( x^2 + y^2 = 1 \), below the plane \( z = 4 \), and above the paraboloid \( z = 1 – x^2 – y^2 \). The density at any point is proportional to its distance from the axis of the cylinder. Find the total mass of \( E \).
  • The total mass as a triple integral is \[ mass=\iiint \limits_E f (x,y,z) dV \]
  • Density at any point in \( E \) is proportional to the distance from \( z \) axis \[ f(x,y,z)= K \sqrt{x^2+y^2} \] where \( K \) is the proportionality constant
  • which in Cylindrican can be re-written as

    \[ f(r,\theta,z)= K r \]
  • The total mass as a triple integral is \[ mass=\iiint \limits_E Kr\;\; dV_{cyl} \]

Example cont'd

  • A solid \( E \) lies within the cylinder \( x^2 + y^2 = 1 \), below the plane \( z = 4 \), and above the paraboloid \( z = 1 – x^2 – y^2 \). The density at any point is proportional to its distance from the axis of the cylinder. Find the total mass of \( E \).
  • Sketch \( E \).
  • Re-define the boundary surfaces in Cylindrical coordinates \[ r=\sqrt{x^2+y^2}\:\: or \:\: r^2=x^2+y^2 \]
  • Cylinder \( x^2 + y^2 = 1 \) becomes \[ r = 1 \] \( \theta \in [0,2\pi] \)
  • Paraboloid \( z= 1-x^2-y^2 \) becomes \( z = 1 – r^2 \)
  • Plane \[ z= 4 \]
  • Set \( E \) in cylindrical coordinates \[ E = \{(r,\theta,z)|\; 0\leq r \leq 1,\; 0 \leq \theta \leq 2\pi,\; 1-r^2 \leq z\leq 4 \} \]
Volume element in cylindrical coords

Courtesy of James Stewart, Calculus: Early transcendentals, 2nd edition

Example cont'd

Volume element in cylindrical coords

Courtesy of James Stewart, Calculus: Early transcendentals, 2nd edition

  • Set up the total mass as a triple integral \[ mass=\int_{0}^{2\pi} \int_0^1 \int_{{\color{purple} {z=1-r^2}} }^{{\color{purple} {z=4}}} ( Kr )\; r\; {\color{purple} {dz}} \;dr\: d\theta \]
  • Integrate with respect to \( z \) first \[ \begin{align} mass&=K \int_{0}^{2\pi} \int_0^1 \int_{{\color{purple} {z=1-r^2}} }^{{\color{purple} {z=4}}} r^2\; {\color{purple} {dz}} \;dr\: d\theta \\ &=K \int_{0}^{2\pi} \int_0^1 \left[r^2\: z\: \right]\bigg\vert_{{\color{purple} {z=1-r^2}} }^{{\color{purple} {z=4}}} \;dr\: d\theta\\ &=K \int_{0}^{2\pi} \int_0^1 r^2 \left[4-(1-r^2) \right] \;dr\: d\theta \end{align} \]
  • Evaluate the double integral in polar coordinates \[ \begin{align} mass& = K \int_{0}^{2\pi} \int_0^1 \left(3r^2+r^4 \right) \;dr\: d\theta \end{align} \]
  • Over a polar rectangle \[ \begin{align} mass&=K \int_{0}^{2\pi} d\theta \int_0^1 \left(3r^2+r^4 \right) \;dr\\ &= K\theta\bigg\vert_{\theta=0}^{\theta=2\pi} \left({r^3+\frac{r^5}{5}} \right)\bigg\vert_{r=0}^{r=1} = K(2\pi)\left( \frac{6}{5}\right) = \frac{12\pi K}{5} \end{align} \]

Example

    Find the volume of a cone of base radius \( R \) and height \( H \) using cylindrical coordinates
Volume element in cylindrical coords

Courtesy of James Stewart, Calculus: Early transcendentals, 2nd edition

  • Volume triple integral \[ V(E)= \iiint \limits_E 1\;dV_{cyl} \]
  • The cone \( E \) has a much simpler description in cylindrical coordinates: \[ E=\left\{(r,\theta,z)|\; 0\leq r \leq R,\; 0 \leq \theta \leq 2\pi,\; \frac{H}{R}r \leq z\leq H \right\} \]
  • used radial symmetry to project to \( rz \)-plane
  • Integrate with respect to \( z \) \[ V(E)= \int_0^{2\pi}\int_0^R \int_{z=\frac{H}{R}r}^{z=H} 1\;r\;dz\;dr\;d\theta \]

Example cont'd

Volume element in cylindrical coords

  • Integrate with respect to \( z \) \[ V(E)= \int_0^{2\pi}\int_0^R \left[\;r\;z \bigg\vert_{z=\frac{H}{R}r}^{z=H}\right] \;dr\;d\theta \]
  • \[ V(E)= \int_0^{2\pi}\int_0^R \left(H r - \frac{H}{R} r^2 \right) \;dr\;d\theta \]
  • Integrate over the polar rectangle \[ \begin{align} V(E)&= \int_0^{2\pi} d\theta \int_0^R \left(H r - \frac{H}{R} r^2 \right)\\ &=(2\pi) \left( H\frac{R^2}{2}-\frac{H}{R}\frac{R^3}{3}\right) \\ &=(2\pi)H\left( \frac{R^2}{2}-\frac{R^2}{3}\right)\\ &=(2\pi)HR^2\left( \frac{1}{2}-\frac{1}{3}\right)\\ &=(2\pi)HR^2\left(\frac{1}{6}\right)=\frac{\pi}{3} HR^2\end{align} \]

Final remarks

  • To change to cylindrical equation use the parametric equations \[ x=r\cos\theta \;\;\; y = r\sin\theta \;\;\;z=z \]
  • Use the volum element in cylindrical coordinates \[ dV_{cyl}= r\;dz\;dr\;d\theta \]
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  • Questions